Question

John will rent a car for a day. The rental company offers two pricing options: Option A and Option B. For each pricing option cost (in dollars ) depends on miles.(1a) If John drives the rental car 120 miles, which option costs less?(1b) How much less does it cost than the other option(2a) for what number of miles driven do the two options cost the same(2b) If John drives more than this amount, which option costs less?

Answer 1

Answer:

1a) If John drives the rental car 120 miles, option B costs less

1b) Option B is is $15 less than option A

2a) The two options cost the same at 60 miles

2b) If John drives more than this amount (50 miles), option B costs less

Explanations:

From the graph provided:

x represents the number of miles driven

y represents the cost

Option B is a constant line graph where y = 50 for all values of x

That is, no matter the number of miles driven in option B, the cost = $50

To find the equation that represents the cost for the number of miles driven in option A, use the equation y = mx + c

where m is the slope and c is the y-intercept

The line touches the y-axis on y = 35, therefore, the y-intercept, c = 35

The slope is calculated using the formula:

$\begin{gathered} m\text{ = }\frac{y_2-y_1}{x_2-x_1} \\ \text{Considering the points (60, 50) and (120, 65)} \\ m\text{ = }\frac{65-50}{120-60} \\ m\text{ = }\frac{15}{60} \\ m\text{ = }0.25 \end{gathered}$

Substituting m = 0.25 and c = 35 into the equation y = mx + c

y = 0.25x + 35

Therefore, the equations representing the two options are:

y = 0.25x + 35 (Option A)

y = 50 (Option B)

1) If John drives the rental car 120 miles, which option costs less?

x = 120 miles

For option A:

y = 0.25(120) + 35

y = 65

Option A costs $65 for 120 miles

Option B costs $50 for 120 miles ( It is a constant graph)

Option B costs less

The difference between the costs of options A and B = $65-$50 = $15

Option B costs $15 less than option A

2) for what number of miles driven do the two options cost the same

For the two options to cost the same, option A must also cost $50

y = 0.25x + 35

y = 50

50 = 0.25x + 35

0.25x = 50 - 35

0.25x = 15

x = 15/0.25

x = 60 miles

The two options cost the same at x = 60 miles

If John drives more than this amount, which option costs less?

If John drives more than 60 miles, option A will cost more than $50, and since option B always costs $50, it will cost less than option A for distances greater 50 miles